Description of KPZ interface growth by stochastic Loewner evolution

TL;DR

This paper explores the connection between the 1D KPZ interface growth model and the stochastic Loewner equation, revealing a correspondence through a nonlinear stochastic process and numerical verification of universality.

Contribution

It establishes a novel link between KPZ interface growth and stochastic Loewner evolution, with analytical and numerical insights into their relationship.

Findings

KPZ height function corresponds to Loewner equation driven by nonlinear stochastic process

Loewner entropy scales as -ln(t/κ), indicating a specific universality behavior

Numerical verification supports the theoretical correspondence and universality in non-equilibrium physics

Abstract

In this study, we investigate the relationship between the one-dimensional (1D) Kardar-Parisi-Zhang (KPZ) equation and the stochastic Loewner equation (SLE), which is a one parameter family of the conformal mappings involving stochasticity. The author shows the correspondence between 1D KPZ equation with height function $h(x,t)=(3t^2x+x^3)/6t$ and Loewner equation driven by a nonlinear stochastic process, wherein the 1D dynamics of interface growth is characterized by Loewner entropy $S_{Loew}\simeq-\ln{t/\kappa}$. These results were numerically verified with discussions in relation to the universality in non-equilibrium statistical physics.